Problem 19 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 9 x+2 y-3 s, \text { if } x=-2, y=5, \text { and } s=-3 $$

Step-by-Step Solution

Answer

The value of the expression is 1.

Step 1: Substitute Given Values

Replace each variable in the expression with the given values: for \( x = -2 \), replace \( x \) with \( -2 \); for \( y = 5 \), replace \( y \) with 5; and for \( s = -3 \), replace \( s \) with \( -3 \). The expression becomes:\[9(-2) + 2(5) - 3(-3)\]

Step 2: Calculate Each Term

Evaluate each part of the expression independently.1. \(9 \times (-2) = -18\)2. \(2 \times 5 = 10\)3. \(3 \times (-3) = -9\)The expression is now:\[-18 + 10 - (-9)\]

Step 3: Simplify the Expression

Add and subtract the evaluated numbers step by step.1. Add the first two terms: \(-18 + 10 = -8\)2. Subtract the third term (note that subtracting a negative is addition): \(-8 - (-9) = -8 + 9 = 1\)

Step 4: Final Result

After performing all the calculations, the final value of the expression is:\[1\]

Key Concepts

Substitution

To solve algebraic expressions like the one given, substitution is the very first step. Substitution means replacing each variable in the expression with the specific value provided. This helps transition an abstract algebraic problem into a numerical one, making it easier to calculate. For instance, in the expression \(9x+2y-3s\), we're given the values \(x = -2\), \(y = 5\), and \(s = -3\). Each variable must be replaced:

• Replace \(x\) with \(-2\)
• Replace \(y\) with \(5\)
• Replace \(s\) with \(-3\)

This substitution transforms our expression into a numerical equation: \[9(-2) + 2(5) - 3(-3)\]. By substituting, we simplify the task, setting the stage for further calculations.

Order of Operations

After substituting the values, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Let's demystify this using the substituted expression \[9(-2) + 2(5) - 3(-3)\]:

• First, carry out the multiplication parts: calculate \(9 \times (-2) = -18\), \(2 \times 5 = 10\), and \(3 \times (-3) = -9\).

This step ensures that each component is evaluated correctly before performing addition or subtraction. The expression is now transformed to \[-18 + 10 - (-9)\]. By following the order strictly, especially in more complex expressions, you avoid mistakes and simplify the computations.

Simplifying Expressions

The final step involves simplifying the expression by performing addition and subtraction in sequence. After the multiplications, the equation becomes \[-18 + 10 - (-9)\]. Calculating step by step:

• Add \(-18 + 10\), which equals \(-8\).
• Note that subtracting a negative number is equivalent to adding: therefore, \(-8 - (-9)\) becomes \(-8 + 9\).
• The result of this is \(1\).

This careful simplification provides the final result. By breaking down the expression thoughtfully, you ensure clarity and accuracy, solving the problem one piece at a time. Simplifying expressions is like solving a puzzle, meticulously connecting the right pieces to reveal the final picture.